May  2016, 21(3): 737-761. doi: 10.3934/dcdsb.2016.21.737

A prey-predator model with migrations and delays

1. 

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's NL, A1C 5S7, Canada

2. 

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

3. 

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's NL, Canada A1C 5S7

Received  January 2015 Revised  September 2015 Published  January 2016

In this paper we propose a prey-predator model in multiple patches through the stage structured maturation time delay with migrations among patches. Focus on the case with two patches, we discuss the existence of equilibrium points and the uniform persistence. In particular, when the maturation times are the same in the patches, we study the local and global attractivity of boundary equilibrium point with general migration function and the local stability of the positive equilibrium with constant migration rate. Numerical simulations are provided to demonstrate the theoretical results, to illustrate the effect of the maturation time, the migration rate on the dynamical behavior of the system.
Citation: Isam Al-Darabsah, Xianhua Tang, Yuan Yuan. A prey-predator model with migrations and delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 737-761. doi: 10.3934/dcdsb.2016.21.737
References:
[1]

K. M. Abadir and J. R. Magnus, Matrix Algebra, Cambridge University Press, 2005. doi: 10.1017/CBO9780511810800.

[2]

A. El Abdllaoui, P. Auger, B. W. Kooi, R. B. Parra and R. Mchich, Effects of density-dependent migrations on stability of a two-patch predator-prey model, Mathematical Biosciences, 210 (2007), 335-354. doi: 10.1016/j.mbs.2007.03.002.

[3]

I. Al-Darabsah and Y. Yuan, Dynamics of a general stage structured n parallel food chains, Proceedings of the International Symposium on Mathematical and Computational Biology, World Scientific, (2014), 63-84.

[4]

V. Andersen, A. Gubanova, P. Nival and T. Ruellet, Zooplankton community during the transition from spring bloom to oligotrophy in the open NW mediterranean and effects of wind events. 2. vertical distributions and migrations, Journal of Plankton Research, 23 (2001), 243-261. doi: 10.1093/plankt/23.3.243.

[5]

R. A. Armstrong, Stable model structures for representing biogeochemical diversity and size spectra in plankton communities, Journal of Plankton research, 21 (1999), 445-464. doi: 10.1093/plankt/21.3.445.

[6]

M. Bandyopadhyay and S. Banerjee, A stage-structured prey-predator model with discrete time delay, Applied Mathematics and Computation, 182 (2006), 1385-1398. doi: 10.1016/j.amc.2006.05.025.

[7]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J Math Anal., 33 (2002), 1144-1165.

[8]

Y. Cai, M. Banerjee, Y. Kang and W. Wang, Spatiotemporal complexity in a predator-prey model with weak Allee effects, Mathematical Biosciences and Engineering, 11 (2014), 1247-1274.

[9]

B. Daia, H. Su and D. Hu, Periodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse, Nonlinear Analysis, 70 (2009), 126-134. doi: 10.1016/j.na.2007.11.036.

[10]

W. Feng, B. Rock and J. Hinson, On a new model of two-patch predator-prey system with migration of both Species, J. Applied Analysis and Computation, 1 (2011), 193-203.

[11]

D. R. French and J. M. Travis, Density-dependent dispersal in host-parasitoid assemblages, Oikos, 95 (2001), 125-135. doi: 10.1034/j.1600-0706.2001.950114.x.

[12]

S. A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, Journal of Mathematical Biology, 49 (2004), 188-200. doi: 10.1007/s00285-004-0278-2.

[13]

A. Hastings and L. Gross, editors, Encyclopedia of Theoretical Ecology, University of California Press, 2012.

[14]

C. S. Holling, The functional response of predator to prey density and its role in mimicry and population regulation, Memoirs of the Entomological Society of Canada, 97 (1965), 5-60.

[15]

Y. Huang, F. Chen and L. Zhong, Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge, Applied Mathematics and Computation, 182 (2006), 672-683. doi: 10.1016/j.amc.2006.04.030.

[16]

V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, 1961.

[17]

V. Jansen, Theoretical Aspects of Metapopulation Dynamics, Ph.D. thesis Leiden University The Netherlands, 1994.

[18]

Y. Kang, S. K. Sasmal, A. R. Bhowmick and J. Chattopadhyay, Dynamics of a predator-prey system with prey subject to Allee effects and disease, Mathematical Biosciences and Engineering, 11 (2014), 877-918. doi: 10.3934/mbe.2014.11.877.

[19]

Y. Kuang, Delay Differential Equation with Application in Population Dynamics, New York: Academic Press, 1993.

[20]

Y. Kuang and Y. Takeuchi, Predator-prey dynamics in models of prey dispersal in two-patch environments, Mathematical Biosciences, 120 (1994), 77-98. doi: 10.1016/0025-5564(94)90038-8.

[21]

M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448. doi: 10.1137/090779322.

[22]

R. Mchich, P. Auger and J. C. Poggiale, Effect of predator density dependent dispersal of prey on stability of a predator-prey system, Mathematical Biosciences, 206 (2007), 343-356. doi: 10.1016/j.mbs.2005.11.005.

[23]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.

[24]

K. Olli, Diel vertical migration of phytoplankton and heterotrophic flagellates in the Gulf of Riga, Journal of Marine Systems, 23 (1999), 145-163. doi: 10.1016/S0924-7963(99)00055-X.

[25]

I. L. Pepper, C. P. Gerba and T. J. Gentry, editors, Environmental Microbiology, Academic Press, 2014.

[26]

P. Pillai, A. Gonzalez and M. Loreau, Evolution of dispersal in a predator-prey metacommunity, The American Naturalist, 179 (2012), 204-216. doi: 10.1086/663674.

[27]

T. S. Shores, Applied Linear Algebra and Matrix Analysis, Springer, 2007. doi: 10.1007/978-0-387-48947-6.

[28]

J. R. Silvester, Determinants of block matrices, Maths Gazette, 84 (2000), 460-467. doi: 10.2307/3620776.

[29]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, 1995.

[30]

H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Analysis, 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2.

[31]

P. D. Srinivasu and I. L. Gayatri, Influence of prey reserve capacity on predator-prey dynamics, Ecological Modelling, 181 (2005), 191-202. doi: 10.1016/j.ecolmodel.2004.06.031.

[32]

X. H. Tang and X. Zou, Global attractivity in a predator-prey system with pure delays, Proceedings of the Edinburgh Mathematical Society, 51 (2008), 495-508. doi: 10.1017/S0013091506000988.

[33]

D. Tilman and P. Kareiva, Spatial Ecology, Princenton University Press, 1997.

[34]

F. Wang, Y. Kuang, C. Ding and S. Zhang, Stability and bifurcation of a stage-structured predator-prey model with both discrete and distributed delays, Chaos, Solitons & Fractals, 46 (2013), 19-27. doi: 10.1016/j.chaos.2012.10.003.

[35]

Z. Wang and X. Q. Zhao, Global dynamics of a time-delayed dengue transmission model, Canadian Appl. Math. Quarterly, 20 (2012), 89-113.

[36]

R. Xu and L. Chen, Persistence and stability for a two-species ratio-dependent predator-prey system with time delay in a two-patch environment, Computers & Mathematics with Applications, 40 (2000), 577-588. doi: 10.1016/S0898-1221(00)00181-4.

[37]

C. Xu, X. Tang, M. Liao and X. He, Bifurcation analysis in a delayed Lokta-Volterra predator-prey model with two delays, Nonlinear Dynamics, 66 (2011), 169-183. doi: 10.1007/s11071-010-9919-8.

[38]

Y. Yuan and J. Bélair, Threshold dynamics in an SEIRS model with latency and temporary immunity, Journal of Mathematical Biology, 69 (2014), 875-904. doi: 10.1007/s00285-013-0720-4.

show all references

References:
[1]

K. M. Abadir and J. R. Magnus, Matrix Algebra, Cambridge University Press, 2005. doi: 10.1017/CBO9780511810800.

[2]

A. El Abdllaoui, P. Auger, B. W. Kooi, R. B. Parra and R. Mchich, Effects of density-dependent migrations on stability of a two-patch predator-prey model, Mathematical Biosciences, 210 (2007), 335-354. doi: 10.1016/j.mbs.2007.03.002.

[3]

I. Al-Darabsah and Y. Yuan, Dynamics of a general stage structured n parallel food chains, Proceedings of the International Symposium on Mathematical and Computational Biology, World Scientific, (2014), 63-84.

[4]

V. Andersen, A. Gubanova, P. Nival and T. Ruellet, Zooplankton community during the transition from spring bloom to oligotrophy in the open NW mediterranean and effects of wind events. 2. vertical distributions and migrations, Journal of Plankton Research, 23 (2001), 243-261. doi: 10.1093/plankt/23.3.243.

[5]

R. A. Armstrong, Stable model structures for representing biogeochemical diversity and size spectra in plankton communities, Journal of Plankton research, 21 (1999), 445-464. doi: 10.1093/plankt/21.3.445.

[6]

M. Bandyopadhyay and S. Banerjee, A stage-structured prey-predator model with discrete time delay, Applied Mathematics and Computation, 182 (2006), 1385-1398. doi: 10.1016/j.amc.2006.05.025.

[7]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J Math Anal., 33 (2002), 1144-1165.

[8]

Y. Cai, M. Banerjee, Y. Kang and W. Wang, Spatiotemporal complexity in a predator-prey model with weak Allee effects, Mathematical Biosciences and Engineering, 11 (2014), 1247-1274.

[9]

B. Daia, H. Su and D. Hu, Periodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse, Nonlinear Analysis, 70 (2009), 126-134. doi: 10.1016/j.na.2007.11.036.

[10]

W. Feng, B. Rock and J. Hinson, On a new model of two-patch predator-prey system with migration of both Species, J. Applied Analysis and Computation, 1 (2011), 193-203.

[11]

D. R. French and J. M. Travis, Density-dependent dispersal in host-parasitoid assemblages, Oikos, 95 (2001), 125-135. doi: 10.1034/j.1600-0706.2001.950114.x.

[12]

S. A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, Journal of Mathematical Biology, 49 (2004), 188-200. doi: 10.1007/s00285-004-0278-2.

[13]

A. Hastings and L. Gross, editors, Encyclopedia of Theoretical Ecology, University of California Press, 2012.

[14]

C. S. Holling, The functional response of predator to prey density and its role in mimicry and population regulation, Memoirs of the Entomological Society of Canada, 97 (1965), 5-60.

[15]

Y. Huang, F. Chen and L. Zhong, Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge, Applied Mathematics and Computation, 182 (2006), 672-683. doi: 10.1016/j.amc.2006.04.030.

[16]

V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, 1961.

[17]

V. Jansen, Theoretical Aspects of Metapopulation Dynamics, Ph.D. thesis Leiden University The Netherlands, 1994.

[18]

Y. Kang, S. K. Sasmal, A. R. Bhowmick and J. Chattopadhyay, Dynamics of a predator-prey system with prey subject to Allee effects and disease, Mathematical Biosciences and Engineering, 11 (2014), 877-918. doi: 10.3934/mbe.2014.11.877.

[19]

Y. Kuang, Delay Differential Equation with Application in Population Dynamics, New York: Academic Press, 1993.

[20]

Y. Kuang and Y. Takeuchi, Predator-prey dynamics in models of prey dispersal in two-patch environments, Mathematical Biosciences, 120 (1994), 77-98. doi: 10.1016/0025-5564(94)90038-8.

[21]

M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448. doi: 10.1137/090779322.

[22]

R. Mchich, P. Auger and J. C. Poggiale, Effect of predator density dependent dispersal of prey on stability of a predator-prey system, Mathematical Biosciences, 206 (2007), 343-356. doi: 10.1016/j.mbs.2005.11.005.

[23]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.

[24]

K. Olli, Diel vertical migration of phytoplankton and heterotrophic flagellates in the Gulf of Riga, Journal of Marine Systems, 23 (1999), 145-163. doi: 10.1016/S0924-7963(99)00055-X.

[25]

I. L. Pepper, C. P. Gerba and T. J. Gentry, editors, Environmental Microbiology, Academic Press, 2014.

[26]

P. Pillai, A. Gonzalez and M. Loreau, Evolution of dispersal in a predator-prey metacommunity, The American Naturalist, 179 (2012), 204-216. doi: 10.1086/663674.

[27]

T. S. Shores, Applied Linear Algebra and Matrix Analysis, Springer, 2007. doi: 10.1007/978-0-387-48947-6.

[28]

J. R. Silvester, Determinants of block matrices, Maths Gazette, 84 (2000), 460-467. doi: 10.2307/3620776.

[29]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, 1995.

[30]

H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Analysis, 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2.

[31]

P. D. Srinivasu and I. L. Gayatri, Influence of prey reserve capacity on predator-prey dynamics, Ecological Modelling, 181 (2005), 191-202. doi: 10.1016/j.ecolmodel.2004.06.031.

[32]

X. H. Tang and X. Zou, Global attractivity in a predator-prey system with pure delays, Proceedings of the Edinburgh Mathematical Society, 51 (2008), 495-508. doi: 10.1017/S0013091506000988.

[33]

D. Tilman and P. Kareiva, Spatial Ecology, Princenton University Press, 1997.

[34]

F. Wang, Y. Kuang, C. Ding and S. Zhang, Stability and bifurcation of a stage-structured predator-prey model with both discrete and distributed delays, Chaos, Solitons & Fractals, 46 (2013), 19-27. doi: 10.1016/j.chaos.2012.10.003.

[35]

Z. Wang and X. Q. Zhao, Global dynamics of a time-delayed dengue transmission model, Canadian Appl. Math. Quarterly, 20 (2012), 89-113.

[36]

R. Xu and L. Chen, Persistence and stability for a two-species ratio-dependent predator-prey system with time delay in a two-patch environment, Computers & Mathematics with Applications, 40 (2000), 577-588. doi: 10.1016/S0898-1221(00)00181-4.

[37]

C. Xu, X. Tang, M. Liao and X. He, Bifurcation analysis in a delayed Lokta-Volterra predator-prey model with two delays, Nonlinear Dynamics, 66 (2011), 169-183. doi: 10.1007/s11071-010-9919-8.

[38]

Y. Yuan and J. Bélair, Threshold dynamics in an SEIRS model with latency and temporary immunity, Journal of Mathematical Biology, 69 (2014), 875-904. doi: 10.1007/s00285-013-0720-4.

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